Maximum height of low-temperature 3D Ising interfaces

Seminar | October 30 | 3:10-4 p.m. | 1011 Evans Hall

 Reza Gheissari, Miller Fellow, U.C. Berkeley

 Department of Statistics

Consider the random surface given by the interface separating the plus and minus phases in a low-temperature Ising model in dimensions $d\geq 3$. Dobrushin (1972) famously showed that in cubes of side-length n the horizontal interface is rigid, exhibiting order one height fluctuations above a fixed point.
We study the large deviations of this interface and obtain a shape theorem for its pillar, conditionally on it reaching an atypically large height. We use this to analyze the law of the maximum height M_n of the interface: we prove that for every beta large,
M_n/log n converges to c(beta), and the sequence M_n - E[M_n] is tight. Moreover, even though this centered sequence does not converge, all its sub-sequential limits satisfy uniform Gumbel tail bounds. Based on joint work with Eyal Lubetzky.